Analysis and Implementation of Approximate Solution Methods to the Equation of Radiative Transfer in Participating Media

Dual Degree Project Final Report

ByV.Narender Reddy10D010003

Under the guidance ofProf. Kowsik Bodi

Department of Aerospace Engineering,Indian Institute of Technology, BombayMay, 2015

Acknowledgement

I am extremely thankful to Prof. Kowsik Bodi for having given me the opportunity to work under him on this topic. He has shown a lot of confidence in me and encouraged me to think on my own. He was always open to whatever ideas I had and at times when I faced difficulties, he continuously mentored me giving me valuable suggestions and new lines to think on. I am also thankful to the Department of Aerospace Engineering, IIT Bombay for providing me with the resources to work on this project.

I would like to thank all of my friends and colleagues for their continuous assistance and backing throughout the project.Finally I would like to thank my Parents for their blessings and continuous motivation, without which reaching at this stage would not have been possible.

Narender Reddy V(10D010003)

Certificate

Abstract

Approximate Methods to solve the Equation of Radiative Transfer in Participating Medium have been used to solve three types of Problems. 1-D Atmospheric Model is built to get Atmospheric Spectrum which is solved by using Runge-Kutta (RK4) Numerical Method. Using method heat transfer rates have been found within the medium between two Surfaces. Finally a two Dimensional Cylindrical Enclosure (with heat Source/s Enclaved in it) problem have been solved for Surface Heat Fluxes Using Monte Carlo Methods. Improvements have been made to these approximation Methods to generalize the problem for many cases which are presented in the report.

Table of Contents

List of Figures & Tables6Nomenclature71.1Motivation81.2Objective of this work81.3Organization of report8Thermal Radiation92.1 Fundamentals of Thermal Radiation92.2 Equation of Radiative Transfer in Participating Media102.3 Exact Solutions112.4 Approximate Methods112.4.2 Discrete Ordinate Method112.4.3 Finite Volume Method122.4.4 Spherical Harmonics132.4.5Monte Carlo Method13Atmosphere 1-D Radiation Model153.1 Problem Statement153.2 Background153.3 Governing Equation153.3.1 Absorption153.3.2 Emission163.4 Approach163.5 Solving Steps163.6 Results & Verification183.6.1 Sun as Source183.6.2 Earth as Source193.6.3 Standard plots & Verification20234.1 Problem Statement234.2 Solution Method234.3 Calculations254.4 Results30Two-Dimensional Cylindrical Enclosure315.1 Problem Statement315.2 Introduction313.3 Description of Monte Carlo Method.323.4 Approach323.6 Results323.6 Error Analysis32References32Appendices34

List of Figures

Nomenclature

Subscripts

Chapter 1Introduction 1.1 Motivation

Energy is Mystery and Thermal Radiation is more Mysterious. In modern era, analysis of radiative heat transfer has been an important field of study. The reasons being Energy Efficiency. With depleting energy sources it has been focus in many engineering areas of heat transfer to carefully design systems to avoid heat losses. High temperature Applications. In case of conduction & convection heat transfer is directly proportional to temperature, but in radiation it is proportional fourth power of temperature. So in higher temperature applications such as rocket nozzles, Jet Engines etc., Radiative heat Transfer should be checked. Environmental Damage. Studying depletion of Ozone Due to pollutant media. To get estimation of temperatures, heat flux etc. at areas where temperatures cannot be measured with equipments. To avoid high cost

1.2 Objective of this work

To study radiative heat transfer 1.3 Organization of report

Initial Part of the report is introduction to equation to Radiative transfer and Literature Survey of major approximate methods to solve it. After that the Atmosphere 1-D Radiation Model is analyzed using Equation of Radiative Transfer. Finally major problem of two Dimensional cylindrical enclosure is analyzed using Monte Carlo Method.

Chapter 2Thermal Radiation 2.1 Fundamentals of Thermal Radiation In Simple terms Thermal Radiation is a mode of heat transfer caused by electromagnetic waves. Anything which has mass and temperature greater than absolute zero continuously emits/absorbs thermal radiation.

Emissive Power: Radiative Heat Flux Emitted from a surface.Total Black Body Emissive Power: Radiative Intensity: Radiative energy flow per unit solid angle and unit area normal to the rays.Radiative Heat Flux:

Transmissivity: where S thickness of gas and And Absorptivity:Extinction Coefficient: Radiative equilibrium means that thermodynamic equilibrium within the medium is achieved by virtue of thermal radiation alone, neglecting conduction and convection.

Diffuse-gray surface: Diffuse word tells us that the directional emissivity and directional absorptivity do not depend on direction. The term gray tells us that the spectral emissivity and absorptivity do not depend on wavelength. They depend on temperature. The diffuse-gray surface absorbs and also emits a fixed fraction of radiation from any direction and at any wavelength.

The scattering phase function describes the probability that a ray from one direction s will be scattered into a certain other direction, S. Linear anisotropic phase function and Rayleigh phase function are two scattering phase function. Isotropic scattering and anisotropic scattering are again two types in linear anisotropic phase function. Isotropic scattering scatters energy equally into all directions. Anisotropic scattering can be further divided into backward and forward scattering. Backward scatters more energy into the backward direction, while forward scattering scatters more energy into the forward directions.Optical thickness is a measure of transparency, and is defined as the negative logarithm of the fraction of radiation that is not scattered or absorbed on a path. If put simply, it shows the proportionality of radiation absorbed or scattered in partially transparent medium ; If an object is very near then optical thickness is almost zero, if distance increases optical depth increases.. If I0 is the intensity of radiation at the source and I is the observed intensity after a given path, then optical depth is defined by the following equation . If >>1, the path in the medium is optically thick. If providing all the necessary classes/functions to deal with shapes/ random spatial distributionsModels -> Models providing interfaces to the physics of the problem, and some other helper classes to ease the modelling of a system containing many zonesMonte Carlo -> Mathematical framework providing the functions for random sampling, simulations and some visualization of the results

GeometryAny system needs to be modelled with shapes, currently we are providing the asic shapes of a rectangle and a circle. Any union/intersection of these shapes can be taken and used to model the geometry of the system

ModelsWorld -- [Zones]Zone -- [Shape, [albedo, absK, energy], type]Photon - [Physics, Interactions]

Monte CarloProduces random photons based on the energy zones and probability distributions, collects results and helps in visualization

3.5 Results Simulations For various Cases are presented below as Solved by Python Code Developed Using Monte Carlo Method.3.5.1 Variable Geometry

2000 Photon generated out of Which 187 are absorbed at surface for a given absorK(0.03) ) and Albedo (0.4.Heat Source at Centre.16 Seconds To Simulate.

Heat Map Of Above problem. Bright is more heat.

9000 Photon generated out of Which 791 are absorbed at surface for a given absorK(0.03) and Albedo (0.4).Heat Source at Centre.80 Seconds To Simulate.

Heat Map of Above problem. Bright is more heat.

Inserting Two Different Heat Sources at Two Different Places With Equal Heat strength.9000 Photons are generated. Took 89 seconds and 1994/9000 are absorbed at wall for 0.4 albedo and o.oo3 absK.

Heat Map of above Figure .Red and Bright signifies higher temperatures.

For 20000 Photons and Different Heat Strengths for Heat Sources (Rec>Circle).

Heat Map For Above Problem.

This Way We can solve for any kind of Geometry at any Strength, at any Places .Some more examples of variable geometries are Given Below.

40000 Simulations at 10 minutes for Different heat Strengths. [Rec>Circles]

Heat Map For Above Figure.3.5.2 Property Changes

Changing Properties (extinction coefficient, absorption Coefficient) of Participating medium, heat Sources (assigning absorption coefficient even to them).We can show results for any geometry but I Fixed a geometry and varied other Media and Surface Properties.

3.5 Number of Photons Required?Now We have Real Working Model For Radiative Heat Transfer Problem For a enclosure which can also handle heat Sources, all we have to do is give inputs to our model which provide approximate solutions to real world systems. One of the biggest Challenge in giving scaled up inputs is deciding how number of photon packets needs to be employed. In Paper written by howell 3.6 Error Analysis

References

Appendices